Optimal. Leaf size=63 \[ \frac {b B x}{e^2}-\frac {(b d-a e) (B d-A e)}{e^3 (d+e x)}-\frac {(2 b B d-A b e-a B e) \log (d+e x)}{e^3} \]
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Rubi [A]
time = 0.04, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {78}
\begin {gather*} -\frac {(b d-a e) (B d-A e)}{e^3 (d+e x)}-\frac {\log (d+e x) (-a B e-A b e+2 b B d)}{e^3}+\frac {b B x}{e^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 78
Rubi steps
\begin {align*} \int \frac {(a+b x) (A+B x)}{(d+e x)^2} \, dx &=\int \left (\frac {b B}{e^2}+\frac {(-b d+a e) (-B d+A e)}{e^2 (d+e x)^2}+\frac {-2 b B d+A b e+a B e}{e^2 (d+e x)}\right ) \, dx\\ &=\frac {b B x}{e^2}-\frac {(b d-a e) (B d-A e)}{e^3 (d+e x)}-\frac {(2 b B d-A b e-a B e) \log (d+e x)}{e^3}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 56, normalized size = 0.89 \begin {gather*} \frac {b B e x-\frac {(b d-a e) (B d-A e)}{d+e x}+(-2 b B d+A b e+a B e) \log (d+e x)}{e^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 70, normalized size = 1.11
method | result | size |
default | \(\frac {b B x}{e^{2}}-\frac {A a \,e^{2}-A b d e -B a d e +B b \,d^{2}}{e^{3} \left (e x +d \right )}+\frac {\left (A b e +B a e -2 B b d \right ) \ln \left (e x +d \right )}{e^{3}}\) | \(70\) |
norman | \(\frac {\frac {B b \,x^{2}}{e}-\frac {A a \,e^{2}-A b d e -B a d e +2 B b \,d^{2}}{e^{3}}}{e x +d}+\frac {\left (A b e +B a e -2 B b d \right ) \ln \left (e x +d \right )}{e^{3}}\) | \(75\) |
risch | \(\frac {b B x}{e^{2}}-\frac {A a}{e \left (e x +d \right )}+\frac {A b d}{e^{2} \left (e x +d \right )}+\frac {B a d}{e^{2} \left (e x +d \right )}-\frac {B b \,d^{2}}{e^{3} \left (e x +d \right )}+\frac {\ln \left (e x +d \right ) A b}{e^{2}}+\frac {\ln \left (e x +d \right ) B a}{e^{2}}-\frac {2 \ln \left (e x +d \right ) B b d}{e^{3}}\) | \(106\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 75, normalized size = 1.19 \begin {gather*} B b x e^{\left (-2\right )} - {\left (2 \, B b d - B a e - A b e\right )} e^{\left (-3\right )} \log \left (x e + d\right ) - \frac {B b d^{2} + A a e^{2} - {\left (B a e + A b e\right )} d}{x e^{4} + d e^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.47, size = 102, normalized size = 1.62 \begin {gather*} -\frac {B b d^{2} - {\left (B b x^{2} - A a\right )} e^{2} - {\left (B b d x + {\left (B a + A b\right )} d\right )} e + {\left (2 \, B b d^{2} - {\left (B a + A b\right )} x e^{2} + {\left (2 \, B b d x - {\left (B a + A b\right )} d\right )} e\right )} \log \left (x e + d\right )}{x e^{4} + d e^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.26, size = 71, normalized size = 1.13 \begin {gather*} \frac {B b x}{e^{2}} + \frac {- A a e^{2} + A b d e + B a d e - B b d^{2}}{d e^{3} + e^{4} x} + \frac {\left (A b e + B a e - 2 B b d\right ) \log {\left (d + e x \right )}}{e^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 2.08, size = 116, normalized size = 1.84 \begin {gather*} {\left (x e + d\right )} B b e^{\left (-3\right )} + {\left (2 \, B b d - B a e - A b e\right )} e^{\left (-3\right )} \log \left (\frac {{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right ) - {\left (\frac {B b d^{2} e}{x e + d} - \frac {B a d e^{2}}{x e + d} - \frac {A b d e^{2}}{x e + d} + \frac {A a e^{3}}{x e + d}\right )} e^{\left (-4\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.09, size = 75, normalized size = 1.19 \begin {gather*} \frac {\ln \left (d+e\,x\right )\,\left (A\,b\,e+B\,a\,e-2\,B\,b\,d\right )}{e^3}-\frac {A\,a\,e^2+B\,b\,d^2-A\,b\,d\,e-B\,a\,d\,e}{e\,\left (x\,e^3+d\,e^2\right )}+\frac {B\,b\,x}{e^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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